Logical Subtraction and Partial Truth

Stephen Yablo gave a very impressive series of lectures here last week. His core insight was that often we can communicate important truths by way of assertions that are literally false. The rough explanation for this is that the assertion, though false, may be "partly true", or "true insofar as it concerns a certain subject matter." For example, 'the number of dragons is zero' may be literally false if numbers do not exist. But it is partly true, i.e. true in what it says about dragons (namely, that there aren't any). If you take the literal meaning, and subtract the claim that there are numbers, the remaining content is wholly true.

Sometimes logical subtraction seems unproblematic, as the subtracted element is "perfectly extricable" from what's being said. Other cases, however, are more problematic. When you subtract the redness from scarlet, what's left? 'Tom [the tomato] is scarlet - Tom is red' does not leave any remainder that we can make sense of. It seems perfectly inextricable. Then there may be inbetween cases, such as Wittgenstein's famous question: "what is left over if I subtract the fact that my arm goes up from the fact that I raise my arm?" We have some grasp of this, but as Jaeger has pointed out, "it is not the case that there is exactly one statement R such that 'R & my arm goes up' is logically equivalent to 'I raise my arm'."

Yablo's solution is to say that "P-Q always exists, but it doesn't always project very far out of the Q-region [of logical space]. Inextricability simply means that it is hard [or impossible] to evaluate P-Q in worlds where Q fails."

Intuitively, we can say that:
(i) P-Q is false iff P adds falsity to Q.
(ii) P-Q is true iff not-P adds falsity to Q.
(iii) If neither P nor its negation adds falsity to Q, then P-Q is undefined (lacks a truth value).

Yablo systematizes our intuitive judgments here by appeal to truthmakers, or the reasons why a proposition is true/false. P "adds falsity" to Q if it is false for a Q-compatible reason, i.e. there is a Q-compatible falsity-maker for P.

Example: Let P = 'The King of France is bald' and Q = 'France has a King'. Then P-Q is false, because of the following Q-compatible falsity-maker for P: the list of all the bald people, none of whom is a King of France. This falsity-maker could exist, and so make P false, even if Q were true and France did have a King. This shows that P is false for reasons over and above the falsity of its presupposition Q.

Here is a bit more technical detail. Let R be a potential candidate for P-Q. Yablo suggests that R is a successful candidate, i.e. R extrapolates P beyond Q, iff the following three conditions are satisfied:
- "Equivalence: within Q, R is true (false) in a world iff P is true (false)." That is, if R = P-Q, then it had better be the case that R&Q = P.
- "Reasons: within Q, a world is R (~R) for the same reasons it is P&Q rather than ~P&Q (...)" This is equivalence as applied to subject matter, rather than just truth conditions.
- "Orthogonality: outside Q, R is true (false) for the same reasons as within." This is the key principle, which really gets at the intuitive notion that we are genuinely extrapolating P rather than simply gerrymandering a proposition that happens to overlap with P in the Q-region (and then becomes wildly different beyond that point).

Example: The material conditional 'if Q then P' fails the orthogonality condition. Outside the Q region, it is true for the simple reason that Q is false, regardless of P. Compare the visual aid below: 'if Q then P' has truth conditions 'P or ~Q', so would include all the white region in R. The gerrymandering is visible in the fact that the R region would then turn a sharp 'corner' once it left the P & Q region. It should instead extrapolate cleanly as shown.